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Consider the Cauchy problem for the $n$-dimensional incompressible Navier-Stokes equations: $\frac{∂}{∂t}u−α△u+(u·∇)u+∇p = f(x, t),$ with the initial condition $u(x, 0) = u_0(x)$ and with the incompressible conditions $∇·u=0,$ $∇·f=0$ and $∇·u_0 = 0.$ The spatial dimension $n ≥ 2.$

Suppose that the initial function $u_0 ∈ L^1(\mathbb{R}^n) ∩ L^2(\mathbb{R}^n)$ and the external
force $f∈L^1(\mathbb{R}^n\times \mathbb{R}^+)∩L^1(\mathbb{R}^+,L^2(\mathbb{R^n})).$ It is well known that there holds the
decay estimate with sharp rate: $(1 + t)^{1+n/2} ∫_{\mathbb{R}^n} |u(x, t)|^2dx ≤ C,$ for all time $t > 0,$ where the dimension $n ≥ 2,$ $C > 0$ is a positive constant, independent
of $u$ and $(x, t).$

The main purpose of this paper is to provide two independent proofs of
the decay estimate with sharp rate, both are complete, systematic, simplified
proofs, under a weaker condition on the external force. The ideas and methods
introduced in this paper may have strong influence on the decay estimates with
sharp rates of the global weak solutions or the global smooth solutions of similar
equations, such as the $n$-dimensional magnetohydrodynamics equations, where
the dimension $n ≥ 2.$

Consider the Cauchy problem for the $n$-dimensional incompressible Navier-Stokes equations: $\frac{∂}{∂t}u−α△u+(u·∇)u+∇p = f(x, t),$ with the initial condition $u(x, 0) = u_0(x)$ and with the incompressible conditions $∇·u=0,$ $∇·f=0$ and $∇·u_0 = 0.$ The spatial dimension $n ≥ 2.$

Suppose that the initial function $u_0 ∈ L^1(\mathbb{R}^n) ∩ L^2(\mathbb{R}^n)$ and the external
force $f∈L^1(\mathbb{R}^n\times \mathbb{R}^+)∩L^1(\mathbb{R}^+,L^2(\mathbb{R^n})).$ It is well known that there holds the
decay estimate with sharp rate: $(1 + t)^{1+n/2} ∫_{\mathbb{R}^n} |u(x, t)|^2dx ≤ C,$ for all time $t > 0,$ where the dimension $n ≥ 2,$ $C > 0$ is a positive constant, independent
of $u$ and $(x, t).$

The main purpose of this paper is to provide two independent proofs of
the decay estimate with sharp rate, both are complete, systematic, simplified
proofs, under a weaker condition on the external force. The ideas and methods
introduced in this paper may have strong influence on the decay estimates with
sharp rates of the global weak solutions or the global smooth solutions of similar
equations, such as the $n$-dimensional magnetohydrodynamics equations, where
the dimension $n ≥ 2.$

*Annals of Applied Mathematics*.

*34*(4). 416-438. doi: